Three-dimensional nonlinear dispersive waves on shear flows

The Green-Naghdi equations describing three-dimensional water waves are considered. Assuming that transverse variations of the flow occur at a much shorter lengthscale than variations along the wave propagation direction, we derive simplified asymptotic equations from the Green-Naghdi model. For steady flows, we show that the approximate model reduces to a one-dimensional Hamiltonian system along each stream line. Exact solutions describing a wide class of free-boundary flows depending on several arbitrary functions of one argument are found. The numerical results showing different patterns of steady three-dimensional waves are presented.